cube roots of unity, or the roots of x3-1=0. And +1, −1, + √−1, or the roots of x1-1=0. √1, are the four fourth roots of unity, 232. It results from the preceding analysis, that the rules for the calculus of radicals, which are exact when applied to absolute numbers, are susceptible of some modifications, when applied to expressions or symbols which are purely algebraic; these modifications are more particularly necessary when applied to imaginary expressions, and are a consequence of what has been said in (Art. 230). For example, the product of a by V-a, by the rule of (Art. 228), would be √ -ax√_a = √+a2. Now, Va is equal to ±a (Art. 230); there is, then, apparent. ly, an uncertainty as to the sign with which a should be affected. Nevertheless, the true answer is -a; for, in order to square √m, it is only necessary to suppress the radical; but the √-ax reduces to (a), and is therefore equal to -a. Again, let it be required to form the product -ax√ --b, by the rule of (Art. 228), we shall have √ -ax√_b = √+ab. Now, Vab=p (Art. 230), p being the arithmetical value of the square root of ab; but I say that the true result should be -p -a or Vab, so long as both the radicals √—a and √—b are con. sidered to be affected with the sign +. For, hence —a=√a. √−1 and √—b=√b. √—1; √=ax √=b= √a. √=1x √ ̄bx √=I= √ ab( √−1)2 = √abx-1=- √ab. Upon this principle we find the different powers of V-I to be, as follows: and ( √−1)'=( √—1)2. ( √—1)2=—1×—1=+1. Again, let it be proposed to determine the product of *√√—a by the V-6 which, from the rule, will be √+ab, and consequently will give the four values (Art. 230). +^√ab, -"√ab, +*√ab. √−1, −ˆ√ab. √ —1. To determine the true product, observe that *√—a=va. √—1, ‘√—b=yb.*√—1. But '√=1x'V=I=('V=1)2={√√. 2 = We will apply the preceding calculus to the verification of the 23-10, that is, as the cube root of 1 (Art. 230). From the formula we have (a+b)3=a3+3a2b+3ab2+b3, -1 + √ = 3√3 (−1)3+3(−1)2. √−3+3(−1).( √−3)2+( √—3)3 233. In extracting the nth root of a quantity a", we have seen that when m is a multiple of n, we should divide the exponent m by n the index of the root; but when m is not divisible by n, in which case the root cannot be extracted algebraically, it has been agreed to indicate this operation by indicating the division of the two exponents. m Hence, "Vaa", from a convention founded upon the rule for the exponents, in the extraction of the roots of monomials. In such expressions, the numerator indicates the power to which the quantity is to be raised, and the denominator, the root to be extracted. In like manner, suppose it is required to divide a" by a". We know that the exponent of the divisor should be subtracted from the am exponent of the dividend, when m>n, which gives :am-n · an But when m<n, in which case the division cannot be effected algebraically, it has been agreed to subtract the exponent of the divisor from that of the dividend. Let p be the absolute difference between Therefore, the expression ar is the symbol of a division which it has been impossible to perform; and its true value is the quotient represented by unity divided by the letter a, affected with the exponent p, taken positively. Thus, The notation of fractional exponents has the advantage of giving an entire form to fractional expressions. From the combination of the extraction of a root, and an impossible division, there results another notation, viz. negative fractional exponents. the radical sign. Hence, a' m a-p, a are conventional expressions, founded up on preceding rules, and equivalent to Va", 1 n 1 am We may therefore substitute the second for the first, or recipro cally. As a is called a to the p power, when p is a positive whole num ists to generalize the word power; but it would, perhaps, be more using the word power only when we wish to designate the product of a number multiplied by itself two or more times. we conclude that any factor may be transferred from the numerator to the denominator, or from the denominator to the numerator, by changing the sign of its exponent. Multiplication of Quantities affected with any Exponents. 234. In order to multiply a by a3, it is only necessary to add the two exponents, and we have or, performing the multiplication by the rule of (Art. 228), Therefore, in order to multiply two monomials affected with any exponents whatever, add together the exponents of the same letter ; this rule is the same as that given in (Art. 41), for quantities affect.. ed with entire exponents. From this rule we will find that a + b + c − × a b + c = a + b + c + ; and Division. 235. To divide one monomial by another when both are affected with any exponent whatever, follow the rule given in Art. 50 for quantities affected with entire and positive exponents; that is, subtract the exponents of the letters in the divisor from the exponents of the same letters in the dividend. For, the exponent of each letter in the quotient must be such, that, added to that of the same letter in the divisor, the sum shall be equal to the exponent of the letter in the dividend; hence the exponent in the quotient is equal to the difference between the exponent in the dividend and that in the divisor. |